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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 663b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 663.a4 | 663b1 | \([1, 1, 1, -539, 4592]\) | \(17319700013617/25857\) | \(25857\) | \([4]\) | \(128\) | \(0.11544\) | \(\Gamma_0(N)\)-optimal |
| 663.a3 | 663b2 | \([1, 1, 1, -544, 4496]\) | \(17806161424897/668584449\) | \(668584449\) | \([2, 4]\) | \(256\) | \(0.46201\) | |
| 663.a2 | 663b3 | \([1, 1, 1, -1389, -14094]\) | \(296380748763217/92608836489\) | \(92608836489\) | \([2, 2]\) | \(512\) | \(0.80858\) | |
| 663.a5 | 663b4 | \([1, 1, 1, 221, 17042]\) | \(1193377118543/124806800313\) | \(-124806800313\) | \([4]\) | \(512\) | \(0.80858\) | |
| 663.a1 | 663b5 | \([1, 1, 1, -20174, -1111138]\) | \(908031902324522977/161726530797\) | \(161726530797\) | \([2]\) | \(1024\) | \(1.1552\) | |
| 663.a6 | 663b6 | \([1, 1, 1, 3876, -89910]\) | \(6439735268725823/7345472585373\) | \(-7345472585373\) | \([2]\) | \(1024\) | \(1.1552\) |
Rank
sage: E.rank()
The elliptic curves in class 663b have rank \(1\).
Complex multiplication
The elliptic curves in class 663b do not have complex multiplication.Modular form 663.2.a.b
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
